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336       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

Since   w* e J$?2(E)   and   (u  H>)* = w* o w* e JS?2(E),   it   follows    that
u o y> e JS?2(E) and that, by (11.5.2) and (15.4.8.1),

(15.4.8.3)                              How|l2^IM|-Nl2-

Hence j?2(E) is a self-adjoint two-sided ideal in the Banach algebra with
involution J$?(E). In general JS?2(E) is not closed in JS?(E) (Problem 12).
For each x e E we can write x =  n#n and IWI2 = X! IJ2- The inter-

n                                               n

pretation given earlier for a Hilbert-Schmidt operator u in terms of /R shows
that the series with general term \n\ - \\u(an)\\ is convergent, and that

(15.4.8.4)                                \\u(x)\\2g \\x\\2-\\u\\},
so that

(15.4.8.5)                                     \\u\\g\\u\\2.

This, together with (15.4.8.2), shows that the norm \\n\\2 on the algebra
^f2(E) satisfies the inequality (15.1.1). Also o$?2(E) contains all continuous
endomorphisms of finite rank. For if u is of finite rank, then its kernel
N = u~~l(G) is closed and of finite codimension in E, hence admits a (finite-
dimensional) orthogonal supplement M in E (6.3.1). If (an) is a Hilbert basis
of E obtained by taking the union of a Hilbert basis of N and a Hilbert basis
of M, then u(a^) = 0 for all but a finite number of indices n; hence u e J5P2(E)-
It follows that 3?2(E) has no unit element, for the unit element (if it existed)
could only be the identity mapping of E, and this is clearly not a Hilbert-
Schmidt operator because E is of infinite dimension.

If u 6 o^2(E), if (an) is a Hilbert basis of E and if un is the continuous
endomorphism of E such that un(ak) = u(ak) for k ^ n and un(ak) = 0 for
k>n, then it is immediate that lim \\u  un\\2 = 0. Hence, in view of

H-+QO

(15.4.8.5), it follows that u is a compact operator (11.2.10). But there exist
compact operators which are not Hilbert-Schmidt operators (Problem 12).
It remains to be shown that <^2(E) is complete with respect to the norm
\\u\\2- Let (un) be a Cauchy sequence with respect to this norm; then by
(15.4.8.5) (un) is also a Cauchy sequence with respect to the norm \\ti\\ on
JS?(E), hence converges to an element v e 3?(E) with respect to this norm. On
the other hand, there exists a real number $ > 0 such that

Iklli- Elem 4 of Section 8.14.)